Distance Between Points (0, A) And (a, 0) On A Coordinate Grid

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Introduction

In the realm of coordinate geometry, determining the distance between two points is a fundamental concept. This article delves into the process of calculating the distance between the points (0, a) and (a, 0) on a coordinate grid. We will explore the application of the distance formula, a cornerstone of coordinate geometry, to arrive at the correct expression representing this distance. Understanding this concept is crucial for various mathematical applications, including geometry, calculus, and linear algebra. This exploration will not only provide a solution to the specific problem at hand but also enhance your understanding of distance calculations in a broader mathematical context. We will break down the distance formula step by step, ensuring clarity and comprehension for readers of all backgrounds. By the end of this article, you will be equipped with the knowledge and skills to confidently calculate distances between points on a coordinate grid.

Understanding the Distance Formula

The distance formula is derived from the Pythagorean theorem, which relates the sides of a right triangle. In a coordinate plane, the distance between two points (x1, y1) and (x2, y2) can be visualized as the hypotenuse of a right triangle. The legs of this triangle are the horizontal and vertical differences between the points' coordinates. The distance formula is expressed as follows:

Distance = √((x2 - x1)² + (y2 - y1)²)

This formula is a powerful tool for calculating the straight-line distance between any two points in a coordinate system. It is essential in various fields, including navigation, engineering, and computer graphics. Understanding the origins of the distance formula in the Pythagorean theorem provides a deeper appreciation for its applicability and versatility. The formula allows us to translate geometric relationships into algebraic expressions, making it a fundamental concept in analytical geometry. In the following sections, we will apply this formula to the specific points (0, a) and (a, 0) to determine the distance between them. By carefully substituting the coordinates into the formula and simplifying the resulting expression, we can arrive at the correct answer.

Applying the Distance Formula to Points (0, a) and (a, 0)

To calculate the distance between the points (0, a) and (a, 0), we will substitute the coordinates into the distance formula. Let (x1, y1) = (0, a) and (x2, y2) = (a, 0). Substituting these values into the formula, we get:

Distance = √((a - 0)² + (0 - a)²)

This substitution sets the stage for simplifying the expression and arriving at the final answer. The careful and accurate substitution of coordinates is a critical step in solving distance problems. It ensures that the correct values are used in the subsequent calculations. The distance formula provides a systematic approach to this process, making it easier to avoid errors. By understanding the formula and practicing its application, we can confidently solve a wide range of distance problems in coordinate geometry. In the next section, we will simplify the expression obtained after substitution to arrive at the final answer.

Simplifying the Expression

Now, let's simplify the expression we obtained after substituting the coordinates into the distance formula:

Distance = √((a - 0)² + (0 - a)²)

First, simplify the terms inside the parentheses:

Distance = √(a² + (-a)²)

Next, square the terms:

Distance = √(a² + a²)

Combine like terms:

Distance = √(2a²)

This simplified expression represents the distance between the points (0, a) and (a, 0). It is a crucial step in arriving at the final answer. The simplification process involves applying basic algebraic principles to reduce the expression to its simplest form. This not only makes the answer easier to understand but also facilitates further calculations if needed. By carefully following the steps of simplification, we can ensure that the final answer is accurate and concise. In the following section, we will analyze the simplified expression and relate it to the given options to determine the correct answer.

Analyzing the Result and Matching the Options

The simplified expression we obtained is √(2a²). Now, let's examine the given options to see which one matches our result. The options are:

  • √(2a²)
  • √(0⁴)
  • √(2a⁴)
  • 0

Comparing our result with the options, we can see that the first option, √(2a²), exactly matches our simplified expression. This confirms that √(2a²) is the correct representation of the distance between the points (0, a) and (a, 0). The other options can be easily ruled out as they do not match the simplified expression. Option √(0⁴) simplifies to 0, which is not the correct distance. Option √(2a⁴) has a different exponent for 'a' and is therefore incorrect. Option 0 represents a zero distance, which is not the case for the given points unless a = 0. Therefore, the correct option is √(2a²). In the next section, we will discuss why the other options are incorrect and further solidify our understanding of the problem.

Why Other Options Are Incorrect

To further solidify our understanding, let's discuss why the other options are incorrect:

  • √(0⁴): This expression simplifies to √0, which equals 0. The distance between (0, a) and (a, 0) is 0 only when a = 0. Otherwise, there is a non-zero distance between the points.
  • √(2a⁴): This expression implies that the term inside the square root has 'a' raised to the power of 4. However, our calculations showed that the correct expression has 'a' raised to the power of 2. This option is incorrect because it does not follow from the correct application of the distance formula and simplification.
  • 0: As mentioned earlier, the distance is 0 only when the two points are the same, which occurs when a = 0. If a is not 0, the distance cannot be 0.

By understanding why these options are incorrect, we reinforce our understanding of the correct solution and the underlying principles of distance calculation. It also highlights the importance of careful application of the distance formula and accurate simplification of the resulting expression. In the concluding section, we will summarize the steps we took to solve the problem and emphasize the key concepts involved.

Conclusion

In conclusion, the distance between the points (0, a) and (a, 0) on a coordinate grid is represented by the expression √(2a²). We arrived at this answer by applying the distance formula, substituting the coordinates of the points, and simplifying the resulting expression. The distance formula, derived from the Pythagorean theorem, is a fundamental tool in coordinate geometry for calculating the straight-line distance between two points. We also discussed why the other options were incorrect, reinforcing the importance of accurate calculations and a thorough understanding of the underlying concepts.

This problem highlights the significance of the distance formula in various mathematical applications. It also underscores the importance of careful substitution, simplification, and analysis of results. By mastering these skills, you can confidently solve a wide range of problems involving distances in coordinate geometry. The ability to calculate distances between points is not only essential in mathematics but also in fields such as physics, engineering, and computer science. Therefore, a strong understanding of this concept is invaluable for anyone pursuing studies or careers in these areas.